In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; in physics, where they give rise to the eigenstates of the quantum harmonic oscillator; in systems theory in connection with nonlinear operations on Gaussian noise. Hermite polynomials were defined by Laplace (1810) though in scarcely recognizable form, and studied in detail by Chebyshev (1859). Chebyshev's work...

== Mehler's formula == Mehler (1866) defined a function and showed, in modernized notation, that it can be expanded in terms of Hermite polynomials H(.) based on weight function exp(−x²) as E ( x , y ) = ∑ n = 0 ∞ ( ρ / 2 ) ...

Wheels are a type of algebra where division is always defined. In particular, division by zero is meaningful. The real numbers can be extended to a wheel, as can any commutative ring. Also the Riemann sphere can be extended to a wheel by adjoining an element 0 / 0 {\displaystyle 0/0} . The Riemann sphere is an extension of the complex plane by an element ∞ {\displaystyle \infty } , where z / 0 = ∞ ...

In mathematics, each closed surface in the sense of geometric topology can be constructed from an even-sided oriented polygon, called a fundamental polygon, by pairwise identification of its edges. This construction can be represented as a string of length 2n of n distinct symbols where each symbol appears twice with exponent either +1 or −1. The exponent −1 signifies that the corresponding edge has the orientation opposing the one of the fundamental polygon. == Examples == Sphere: A A − 1 ...